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Notre Dame Journal of Formal LogicVolume 25, Number 4, October 1984

Generalized Quantifiers and the

Square of Opposition

MARK BROWN*

/ Introduction Work by Rescher and Gallagher [11], and more recently byGeach [5], Peterson [9], Thompson [12], and Peterson and Carnes [10], providesstrong evidence that syllogistic logic and the method of Venn diagrams can beextended to accommodate sentences of such forms as

(1.1) Almost allS are PMost S are PMany S are PFew S are P.

This suggests that we should be able to modify the first-order predicate calcu-lus to provide for renderings of such sentences, and of arguments involving suchsentences. Recently Bar wise and Cooper [2] have given the formal syntax andsemantics for a family of such modifications. In this family of languages, as inthe other works cited above, Almost all', 'most', 'many', and 'few' are treatedas quantifiers analogous in certain important respects to 'all' and 'some'. ButBarwise and Cooper, unlike the other authors cited, do not treat these as meread hoc additions to our stock of quantifiers. Instead they treat them as merelya few from among an indefinitely large class of quantifiers, including 'the' (inits use in definite descriptions), 'both', 'at least seven', 'infinitely many', 'all butthree', 'with at most three exceptions', and a host of others.

The generality of the treatment given by Barwise and Cooper suggests thatif we are to continue to explore the logical properties of generalized quantifiersas viewed from a perspective like that of traditional logic then we should nolonger be content to do so piecemeal. There are, in fact, strong reasons forstudying generalized quantifiers from a traditional perspective, for (as we shall

*I wish to thank the anonymous referee, who made several useful suggestions, and whopointed out an important mistake in an earlier draft of this paper.

Received March 22, 1983; revised July 27, 1983

304 MARK BROWN

see) the syntax and semantics that generalized quantifiers seem to demand re-sist introduction of the kind of instantiation and generalization rules that we maytake as characteristic of (natural deduction treatments of) the standard modernpredicate calculus, but do lend themselves to rules of immediate inference andof syllogistic inference like those of traditional logic. This is not to say thatgeneralized quantifiers should be studied exclusively from the traditional per-spective, of course, but only that such study may well yield some insights, par-ticularly about the relation of logic to ordinary language, that we would notreadily discover by other means. In this paper I initiate a general study of quan-tifiers from a traditional perspective by studying rules of immediate inference,and associated squares of opposition, appropriate to various classes of gener-alized quantifiers.

2 Syntax Consider the following sentence:

(2.1) All men are greedy.

For purposes of rendering this sentence in the notation of standard logic, we areaccustomed to paraphrase 2.1 as:

(2.2) Take any object x: if x is a man then x is greedy.

We then render this as:

(2.3) (vx)[MxDGx].

Now let us try to parallel this analysis with an analysis of:

(2.4) Most men are greedy.

To effect a similar analysis, we would have to find some way in which to com-plete

(2.5) Take most objects x:...

so as to provide a reasonable paraphrase of 2.4; or, at the very least, we wouldhave to find or devise some connective to put in place of '?' in

(2.6) (μx) [Mx ? Gx].

But there seems to be no satisfactory way of completing 2.5, and none of thesixteen possible truth-functional connectives gives satisfactory results when putin place of '? ' .

One way of describing the difficulty is to say that, in claiming that mostmen are greedy, we are not talking about most objects, but only about mostmen. This calls to mind the fact that we do have an alternative way of render-ing 2.1, using relativized or restricted quantification:

(2.7) {Vx:Mx)[Gx].

If we adopt this, rather than 2.3, as our model, we can render 2.4 as

(2.8) (μx:Mx)[Gx]

and sidestep the need to find some connective to replace the '?' in 2.6. We know,

GENERALIZED QUANTIFIERS 305

of course, that in the case of the universal and existential quantifiers the use ofrestricted quantification is eliminable in favor of unrestricted quantification, andit is usual to think of unrestricted quantification as more fundamental. How-ever, we can also eliminate unrestricted universal and existential quantificationin favor of their restricted versions, and thus it is open to us to think ofrelativized quantification as more fundamental. It has been argued (for ex-ample by McCawley [7], pp. 118-123) that in the syntax of natural language therestricted form is fundamental. If that is so, then perhaps the availability of anequivalent unrestricted version may be a misleadingly special feature of 'all' and'some'. Perhaps for most quantifiers, 'most' included, only a restricted formis available. It is fundamental both to the syntax and to the semantics of the ap-proach taken by Barwise and Cooper to assume that all quantification is fun-damentally restricted quantification.

It will be useful to introduce some terminology for discussing the syntaxof such expressions as 2.8. There are several component expressions for whichwe may wish to have designations:

(2.9a) μ(2.9b) μx(2.9c) (μx:Mx)(2.9d) Mx(2.9e) Gx.

Of these, each of 2.9a-c might lay claim to the title Quantifier', but it wouldbe needlessly confusing to award the title to all three. Each choice of one of thethree as rightful owner of the title has disadvantages, but on balance it seemsto me best to reserve the term 'quantifier' for 2.9a. (In this judgment I differfrom Barwise and Cooper.) I shall use the following terminology:

(2.10a) quantifier(2.10b) prefix (of the quantifier phrase)(2.10c) quantifier phrase(2.10d) sortal phrase(2.10e) matrix (of the formula).

This terminology has the advantage of preserving the pattern established by ex-isting use of the terms 'noun', 'noun phrase', 'preposition', 'prepositionalphrase', and the like. In somewhat less formal usage, I shall sometimes refer toMx as the subject, and to Gx as the predicate of 2.8, and refer to the subjectand predicate as the terms of a formula.

As the formal language to be studied, I take a language GQ for general-ized quantification that contains denumerably many quantifier constants,denumerably many one-place predicate constants, denumerably many TI-placerelation constants (for each n greater than one), denumerably many individualconstants, denumerably many individual variables, the usual sentential connec-tives, three operator constants (suffixed superscript prime and asterisk, andprefixed superscript minus sign), and five punctuation marks (left and rightparentheses, left and right brackets, and the colon). We give a recursive defi-nition of quantifiers:

306 MARK BROWN

(2.11) If Q is any quantifier constant, then Q is a quantifier;if Q is any quantifier, then (Q'), (Q*), and (~Q) are quantifiers.

Normally the parentheses can be omitted from the designation of complex quan-tifiers, as we shall see. In the recursive definition of formulas, only the last clauseis novel, compared to standard logic:

(2.12) If P is any one-place predicate constant, and t is any individual constantor individual variable, then Pt is a formula;if R is any Λ-place relation constant and t\,..., tn are any n individualconstants or variables (not necessarily distinct), then Rt\... tn is aformula;if A and B are any formulas, then ~A, (A&B), (AvB)9 (ADB),and (A=B) are formulas;if Q is any quantifier, x is any individual variable, and A and B are anyformulas, then (Qx:A)[B] is a formula.

I will wait until after the presentation of the semantics for GQ to give rules forthe system.

3 Semantics If we wish to elaborate on a sentence such as

(3.1) Most senators are males

one way in which to do so is this:

(3.2) The senators that are males constitute most of the senators.

This kind of paraphrase is the key to the formal semantics I shall offer for quan-tifiers.1 The idea it suggests, recast in a way that foreshadows the set-theoreticlanguage we are accustomed to use for truth-conditions, is this: there is no onesenator, or even any one group of senators, that is meant when we speak of mostsenators (we don't mean the alphabetically first fifty-one of them, for example);rather, there are various collections of senators any one of which would countas constituting most senators, and it happens that the group of male senatorsis one of them. In our models for the formal system GQ, we must assign valuesnot only to the individual, predicate, and relational constants of the language,but also to the quantifier constants. The value assigned to a quantifier whoseintended interpretation is most, for example, should do the job of telling uswhich groups of senators do, and which do not, count as most of them. Simi-larly, it must tell us which subsets of the set of prime numbers do, and whichdo not, count as containing most prime numbers. In general, for each set thatmight serve as the extension of a predicate the value assigned to the quantifiershould enable us to tell which of that set's subsets would and which would notcount as containing most of the elements of that set. The sort of thing that cando this job is a function which, given any subset of the domain as its argument,returns as its value an appropriate collection of the subsets of that argument.I will call such a function a quantificational function, and we can define quan-tificational functions formally as follows:

GENERALIZED QUANTIFIERS 307

(3.3) F is a quantificational function for a set D iff F: (PD -+ (?(?D and foreach EQD,F(E)<^β>E.

Given any quantificational function F on A we can define three other relatedquantificational functions F', F*, and ~F on D, as follows:

(3.4) for each EQDF'{E) = {EO^E:E-Eoe F(E)}F*(E) = {EoςzE:E-Eo£F(E)}-F(E) = {E0^E:E0£F(E)}.

We can now define a model for GQ to be an ordered pair (D, V) consisting ofa nonempty domain D and a valuation V satisfying the following conditions:

(3.5a) if t is any individual constant or individual variable, V(t) SD(3.5b) if P is any one-place predicate constant, V(P) <Ξ D(3.5c) if R is any Λ-place relation constant, V(R) c Dn

(3.5d) if <2 is any quantifier constant, V(Q) is a quantificational functiononZ)

(3.5e) if Q is any quantifier, with F= V(Q), then K(Q') =F', V(Q*) = F*,and K("Q) = " F .

We require one auxiliary notion before we can state the truth-conditionsfor formulas in GQ.

(3.6) If M = <£>, V) is any model, with e E D and x an individual variable, thenby M[e/x] we mean the model (D, V[e/x)) whose domain is the sameand whose valuation agrees in value with V for every argument except(possibly) x, for which V[e/x] (x) = e.

If A is any formula of GQ and M is any model for GQ, we abbreviate the claimthat A is true in the model M by writing that Mt=A, and define this by the fol-lowing truth-conditions:

(3.7a) if A is Pt, where P is any one-place predicate constant and t is anyindividual constant or individual variable, M\=^A iff V(ί) E V(P)

(3.7b) if A is Rtχ...tn9 where R is any w-place relation constant and tu.. .,tn

are any n individual constants or individual variables, M\=A iff

(Kω κω>ew(3.7c) if A is ~ £ for some formula B, Mt=A iff it is false that M^B(3.7d) if A is (£&C) for some formulas B and C, Λf ̂ A iff Mt= £ and

(3.7e) if ,4 is (£vC) for some formulas 5 and C, Mt=v4 iff M t = 5 orM\=C

(3.7f) if ̂ is (B D C) for some formulas B and C, Mι= ,4 iff either it is falsethat Mt= £ or it is true that M\=C

(3.7g) if y4 is (B = C) for some formulas B and C,Mt= A iff M t ( 5 3 C )a n d M t ( C 3 5)

(3.7h) if >4 is (Qx:B)[C] for some quantifier Q, some variable x, and someformulas B and C, Λf *= A iff JD(5) Π D(C) E F(D(B)), where £>(£)is {eeD:M[e/x] ^ B} and £>(C) is {eeD:M[e/x] t= C} and F is

K(G)

308 MARK BROWN

Perhaps it would be helpful to look at an example or two, in order to seehow the semantics works. Suppose S and P are one-place predicate constantsand M = (D, V) is a model in which V assigns the set of all senators in D as thevalue of 5, and the set of all procrastinators in D as the value of P. Then Sx willbe true in M iff the value assigned to x is a senator. D(Sx) will be just the setof all elements of D such that if they had been the value assigned to x, Sx wouldhave been true in M. In short, D(Sx) will be just V(S), i.e., the set of senators.Similarly D(Px) will be the set of values of x for which Px would be true in M,i.e., the procrastinators, i.e., V(P). D(Sx) Π D(Px) is the set of senators whoare procrastinators. Suppose Q is a quantifier, with V(Q) = F, and suppose Fis such that, for each £ c £>, F(E) = {£}. Then F(D(Sx))9 i.e., F(V(S))9 willbe just {V(S)}. So (Qx:Sx) [Px] will be true in M iff the set of senators whoare procrastinators is the set of all senators, namely V(S), i.e., iff all senatorsare procrastinators. F'{ V{S)) will turn out to be just {0}, so (Q'x:Sx) [Px] willbe true in Miff no senators are procrastinators. F*(V(S)) is (?V(S) — {0}, so(Q*x:Sx)[Px] will be true in M iff some senators are procrastinators.~F(V(S)) is (PV(S) - {V(S)}9 and (~Qx:Sx)[Px] will be true in Miff somesenators are not procrastinators.

The quantificational function F just considered corresponds to the universalquantifier. If a quantificational function F\ is such that, for each E^D,FX(E) = {E0^E:\E0\ > \E-E0\}> and if V(Q{) =FU then (Qιx:Sx)[Px] willbe true in M iff the number of senators who are procrastinators exceeds thenumber who are not, i.e., iff most senators are procrastinators. Suppose F2 issuch that, for each E^D, F2(E) = {E} if \E\ = 1, and F2(E) = 0 otherwise;then if V(Q2) = F2 we will find that (Q2x:Sx) [Px] is true in Miff there is ex-actly one senator and he or she is a procrastinator, i.e., iff the senator is aprocrastinator.

By the definitions of/7', F*9 and ~F(for given quantificational functionF) that were given in 3.4, we find that in general (Ff)' = (F*) * = ~(~F) = F.Moreover: (FT = (F*)' = ~F\ ~{F') = (~F)'=F*; and ~(F*) = ('F)*=F'.Corresponding results hold for quantifiers, by 3.5e. For example, since(Ff)* = ~F, we find that for every M, S, and P, Mt= (((QT)x S)[P] iffMt= (C~Q)x:S)[P]. As a consequence we can dispense with the parenthesespreviously required by 2.11. Corresponding syntactic results can be shown to fol-low from the rules presented in the next section.

In view of the close semantic and syntactic connections between a givenquantifier Q and the associated quantifiers Q\ Q*, and ~Q9 I call them cognatequantifiers, and describe each as a cognate of (itself and) the others. I refer toQ' as the obverse of Q, to Q* as the dual of Q, and to ~Q as the contradictoryof β.

4 General quantifier rules One of the interesting problems in the theory ofgeneralized quantifiers is that of formulating general rules of inference applicableto all quantifiers. At first glance it might appear that there could be none. Howcould any rule, even a wildly gerrymandered one, be equally applicable to 'alΓ,'some', 'most', and 'the'? But there are a few, impressions to the contrary not-withstanding. One such rule (or pair of rules, depending on how you choose to

GENERALIZED QUANTIFIERS 309

count) arises from the fact that in clause 3.7h of the truth-conditions the truthvalue of the quantified formula is made to depend only on the extensions of theterms of the formula. As a consequence, for any quantifier Q, any variable x,and any formulas A, Λu A2, B, Bu and B2 we have the following rule(s):

(4.1a) v-A\ = A2 Rule E (extensionality)

:.(Qx-*Ax)[B] = (Qx:A2)[B]

(4.1b) \-Bx=B2

:.{Qx'A)[Bx\ = {Qx:A)[B2\.

That is, provably equivalent sortal phrases may be substituted for one another,and so may provably equivalent matrices.

Another rule arises from the fact that the extension of the matrix affectsthe truth of the quantified formula only to the extent that it intersects the ex-tension of the sortal phrase. Thus the set-theoretic equality D{B) ΠD(C) =D{B) Π (D(B) ΠD{C)) gives rise to the following rule, for any quantifier Q,any variable x, and any formulas B and C:

(4.2) (Qx:B)[C] :: (Qx:B)[(B&C)]. Rule A (absorption)

Here the quadruple dot (according to a notation first introduced by Kahane [6])indicates that the expressions on either side may be substituted for one anothersalve veritate in any (extensional) context. I call this the rule of absorption byanalogy with the propositional calculus rule so designated in Copi [4]. This rulecorresponds to the fact that we can paraphrase the claim that most men aregreedy as the claim that most men are greedy men, and similarly with 'all', 'few','the', etc.

A third rule arises from the fact that, barring collision of bound variables,one bound variable may be substituted uniformly for another in any context.Thus, for any quantifier Q9 any variables x and y, and any formulas Ax and Bx,if Ax and Bx contain no occurrences of y, and if Ay and By are the results ofsubstituting occurrences of y for all free occurrences of x in Ax and Bx respec-tively, then:

(4.3) (Qx:Ax) [Bx] :: (Qy:Ay) [By]. Rule B (bound variables)

Three more rules, which could have been recast as definitions, correspondto the three provisions of clause 3.5e in the definition of a model. For any quan-tifier Q, any variable x9 and any formulas A and B:

(4.4a) (Q'x:A)[B] :: (Qx:A)[~B] Rule O (obversion)(4.4b) (Q'x:A)[~B]::(Qx:A)[B](4.4c) CQx:A)[B] :: {Q*x:A)[~B](4.4d) CQx' A)[~B]::(Q*x:A)[B](4.5a) (Q*x:A)[B] :: ~(Qx:A)[~B] Rule D (duality)(4.5b) (Q*x:A)[~B]::~(Qx:A)[B](4.5c) CQx:A)[B]::~(Q'x:A)[~B](4.5d) CQx:A)[~B]::~(Q'x:A)[B](4.6a) CQx:A)[B] :: ~(Qx:A)[B] Rule C (contradiction)(4.6b) (Q*x:A)[B]::~(Q'x:A)[B].

310 MARK BROWN

It would of course suffice to take one version of each of these rules; the remain-ing versions are merely useful equivalents.

Other rules can be derived from the rules given so far. For example,using Rules E and A, we can easily show that:

(4.7) (Qχ:A)lB]::(Qx:A)[AsB] since \-(A &B) = (A& (A = B))(4.8) (Qx:~A)[B] :: (Qx:~A)[~(A = B)) since v-(~A=B) =

~{A=B)(4.9) (Qχ:A)[B] :: (Qx:A)[ADB] since \-(A&B) = (A&(ADB))(4.10) (Qx:A)[A] :: (Qx:A)[A v B] since \-A s {A & {A vB))(4.11) (Qx:A)[A] :: (Qx:A)[A v ~A] (corollary to 4.10)(4.12) (Qx:A)[A] :: (Qx:A)[Bv~B] (corollary to 4.11)(4.13) (Qx:A&~A)[B] :: (Qx:A & ~A)[C] since \-((A & -A) &B) =

((A&~A)&C)(4.14) h-(ADB)

(Qx:A)[B]

:.(Qx:A)[A] since if ϊ-(ADB), then \-(A&B)=A

It is an open question2 whether there are any rules independent of 4.1-4.6and sound under the semantics for GQ. The semantics for GQ could be augmen-ted by special restrictions on the quantificational functions to be associated withparticular quantifier constants. If that were done some of the quantifier con-stants could be treated as logical constants, and would have special rules as-sociated with them. This would be a reasonable treatment for 'all', 'the', 'both','the n\ 'at least n\ 'at most n\ 'exactly n\ and even 'most', to name only afew. But for many quantifier expressions in natural language, such as 'many','several', and 'almost all', this would be an unreasonable treatment. We shouldno more expect to be able to give precise semantics for 'several' than we expectto give semantics for 'red' that will distinguish its logical role from that of allother color words. In GQ all quantifier constants are treated as nonlogical con-stants, and none have special semantic restrictions or special rules of inferenceassociated with them. This enables us to use GQ to study the general proper-ties of quantifiers.

5 Relations among categorical schemata Given any quantifier Q, we maydistinguish thirty-two cognate categorical schemata whose logical relationshipsto one another are of interest, especially in connection with the study of squaresof opposition. These are exhibited in Table 1. For the sake of brevity, the vari-able of quantification, and the colon, are omitted from the specification of theschemata. The variable of quantification can be assumed to be the samethroughout.

Table 1: Cognate

(QX)[Y] (QX)l~Y\ (Q~X)IY] (Q~X)l~Y)(Q'X)IY] (Q'X)l-Y] (Q'~X)IY] (Q'~X)l~Y](Q*X)IY] (Q*X)l~Y] (Q*~X)IY] (Q*~X)l~Y]CQX)[Y] CQX)l~Y] CQ~X)IY] CQ~X)l~Y]

GENERALIZED QUANTIFIERS 311

Each of the forms in the second row is equivalent to one of the forms inthe first row by obversion. For example, (Q'X)[Y] is equivalent to(QX)[~Y]. Each of the forms in the third row is equivalent to the negation ofthe form immediately above it in the second row, and therefore to the negationof one of the forms in the first row. Each of the forms in the bottom row isequivalent to the negation of the form directly above it in the top row. Thus allfurther logical relations among the thirty-two cognate categorical schemata maybe reduced to relations among the eight schemata in the top row and, indeed,to relations between (QX) [ Y] and its cognate schemata (including itself) in thetop row.

If A and B are any two of the cognate categorical schemata from Table 1(not necessarily distinct, and not necessarily from the top row) then it is of in-terest to examine, for any given quantifier Q, which of fifteen possible relationsmay obtain between them. The fifteen relations of interest are given in Table 2.

These relations among schemata are not independent of one another, as isindicated in Figure 1 in which arrows between relations indicate that having therelation at the base of the arrow entails having the relation at its tip.

Some of the nominally possible relations between cognate categorical sche-mata A and B are in fact not possible for certain choices of A and B. For ex-ample, by Rule C it is impossible to have {QX)[ Y] = (~QX)[Y],or to have(Q'X) IY] Ξ (Q*X) IY] But in addition to cases of this sort, we find it is im-possible to have (QX)[Y] +• (QX)[~Y] For suppose otherwise: then byappropriate instantiation in the definition we can get both

(QS)[P]&~(QS)[~P]

and also

{QS)[~P)&~(QS)[~~P].

But this is a contradiction. By similar reasoning we find that for no two sche-mata A and B chosen from the same row in Table 1 do we ever have eitherA-hB OΪ A^tB.

Moreover, some of the nominally distinct relations between cognate sche-mata turn out to be equivalent for certain choices of schemata A and B. Thus,for example:

VXY[(QX)[Y]D(QX)[~Y]]iff VXY[{QX)[~Y]D(QX)[~~Y)}iff vXY[(QX)[~Y]D (QX)[ Y] ] (by Rule E)iff VXY[(QX)IY] = (QX)1~Y]].

Categorical Schemata

(QY)[X] (QY)[~X] (Q~Y)[X] (Q~Y)[~X](QΎ)IX) (QΎ)[~X] (Q'-Y)IX] (Q'~Y)l~X](QΎ)IX] (Q*Y)l~X] (Q*~Y)[X) (Q*~Y)[~X)CQY)[X] ΓQY)[~X] CQ~Y)IX] CQ~Y)l~X]

312 MARK BROWN

Table 2. Possible Logical Relations Between Cognate Categorical Schemata

PossibleCombinations

A: t t f fSymbol Name Definition B: t f t f

ALB loose relationship VXY[Av ~A] x x x xASB subcontrariety vXY[AvB] x x xA <- B super alternation VXY[B DA] x x xAΊXB pre-truth VXY[A\ x xA -+ B subalternation VXY[A D B] x x xAΎ2B post-truth VXY[B] x xA «-> B equivalence VXY[A = B] x xAΎB co-truth VXY[A&B] x^C,β contrariety vXY[~Av~B] x x x^4—5 contradictoriness VXY[A = ~B] x x^ F 2 5 post-falsity VXY[~B] x xyl +• B anti-subalternation VXY[A & ~B] x^ F J J S pre-falsity v^yf-^l] x xA^tB anti-superalternation VXY[~A &B] xAFB co-falsity VXY[~A & ~B] x

ΘFigure 1. Connections among relations.

GENERALIZED QUANTIFIERS 313

It follows that

(QX)IY]->(QX)[~Y]iff (QX)lY]<r-(QX)[~γ]

iff (QX)IY]~(QX)[~Y].

By similar argument it is easily shown that, for any schemata A and B from thesame row of Table 1, A -* B iff A <-B iff A «-5; and v4 T! £ iff ,4 T 2 £ iff.4 T £ ; and ,4 F! 5 iff A Y2B iff ,4 F £ .

Table 3 shows which relations are possible between cognate schemata withthe same quantifier, i.e., between schemata from the same row of Table 1. Forthis purpose it suffices to show which relations are possible between the schema(QX) [ Y] and its cognates from the top row of Table 1. An entry in the tableother than '0' or 'ft' designates a particular quantifier (the semantics for whichis given following the table) such that if Q is interpreted as that quantifier thepair of schemata at the head of the entry's column will bear the relation cor-responding to the entry's row in Table 3, and no stronger relation. The relationsare listed in monotonically decreasing order of strength, as shown in Figure 1.A <0> indicates that the pair of schemata in question cannot bear the given re-lation to one another, and 'ft' indicates that the pair of schemata in question canonly bear the given relation when they bear some stronger relation given above.Rows (for +• and <+) which would only contain '0's, and rows (for -», <-, T 1 ?

T2, Έu and F2) which would only contain 'ft's have been omitted. For the sakeof brevity, punctuation has been omitted from the representations of the sche-mata at the head of columns. (The '0's in columns 2 and 6-8, and only those,depend on the use of Rule A.)

The various quantifiers designated in Table 3 as illustrations of the possi-bility of the relations exhibited can be defined semantically by the condition$that, in any model M- (D, V):

V(QF) = {(E,0):E^D}V(Qτ) = {(E,(?E):E^D}

Table 3. Distinct Possible Logical Relations Between CognateCategorical Schemata with the Same Quantifier

1 2 3 4 5 6 7 8A QXY QXY QXY QXY QXY QXY QXY QXYB QXY QX~Y Q~XY Q~X~Y QYX QY~X Q~YX Q~Y~X

AΈB QF QF QF QF QF QF QF QF

AΎB Qτ Qτ Qτ Qτ Qτ Qτ Qτ Qτ

A~B Qv Qa Qβ Qβ Qi QΎ Qy Q v

A-B 0 0 Qa Qa 0 0 0 0A SB || Q: Qa Qa Qh Qh Qδ QίACB ii a Q« Q: -& Q; Q; Q;ALB II Q v β v Q v β v Q v Q v Q3

314 MARK BROWN

V(Qv) = {(E9{E}):EςiD}V(Qi) ={(E,(?E-{0}):E^D}V(Qa) = {(E,(?E): ECD} U {(D,0)}V(Qβ) = {<£,0>: 0C£C£>}U{<0,(P0>, <£>,(*>#>}V(QΊ) = {<£,(?£>: E^D& \D\ > 2} U {<£,0>: £gZ>& |Z?| < 2}V(Qδ) = {<£,(?£>: ^ C D } U {<£>,{£>}>}K(βe) = {<£,{£}>: £ c D & \E\ = 1} U {<£,0>: £<Ξ D& | £ | * 1}> m ) ={<^,(P^>: F(*)e£c;£>}U{<£,0>: V(a)^E^D},

where # is some individual constant of the language.

It is easily verified that each of these is, as defined, a quantifier. In par-ticular, Qv and Q3 are the ordinary universal and existential quantifiers and Qe

is the quantifier corresponding to Russellian definite descriptions. QF and Qτ

are the degenerate quantifiers that produce only false and true formulas, respec-tively. The remaining quantifiers are more exotic and bear only uninterestinglycomplex relations to expressions common in ordinary discourse. It is also sim-ple to verify that each of these quantifiers induces relations among schemata asindicated in the table, and induces no stronger relation between the same sche-mata. For example, as is indicated in column 5, row 5, (QδX)[Y] and(QδY)[X] are subcontraries, because: it is possible for (QδX)[Y] to be false;but that can occur only when the extension of X is the whole of the domain Dbut the extension of Y is not, and in such circumstances (QδY)[X\ is true;moreover it is possible for both to be true, e.g., when neither Jfnor Y has thewhole domain D as its extension. By Rule C, (QδX) [ Y] s ~(~QδX) [ Y] and(QδY)lX) Ξ ~CQδY)lX], so from the fact that (QδX)[Y] and (QδY)[X)are subcontraries we can immediately infer that (~QδX)[Y] and {~QδY)[X]are contraries. The proofs for other quantifiers are similar.

To show the impossibility of those relations indicated as such by a '0' inTable 3, it is convenient to let F (as distinguished from F, which designates arelation between categorical schemata) be any logically false formula, say(Px& ~Px)y and then let T be ~F. To show that for no quantifier Q can(ζλY)[Γ] and (QX)[~Y] be contradictories, assume otherwise. Then for someQ we must have VXY[(QX)[Y] = ~(QX)[~Y]]. In particular, then,(QF)[T] = ~(QF)[~T\. But by Rule A (QF)[T] = (QF)[F&T], and byRule E (QF)[F&T] s (QF)[F]. On the other hand ~(QF)[~T] = ~{QF)[F],So (QF)[F] Ξ ~(QF)[F]9 which is a contradiction. The other impossibilityproofs are similar.

It should be noted that Qτ and QF are not merely examples of quantifiersthat induce the relations of co-truth and co-falsity, respectively. They are essen-tially the only such quantifiers. For if Q is such that (QX) [ Y] is co-true withsome other schema, then it is such that (QX) [ Y] is true for all X and F, andis therefore equivalent to (QTX)[Y]; similarly for QF.

6 Relations between cognate quantifiers If R is any of the fifteen relationsdefined in Table 2, and if Qx and Q2 are any cognate quantifiers, then letQx R Q2 abbreviate the claim that {QXX) [ Y] R (Q2X) IY] - We can then thinkof this either as expressing a relation between certain cognate schemata or as ex-pressing a relation directly between the two cognate quantifiers. The six possi-

GENERALIZED QUANTIFIERS 315

ble relations between Q and Q' are, in the light of Rule O, given in column 2of Table 3. Using Rules O, D, and C we can verify that the relation between Qand <2', or rather the strongest relation that holds between them, determines thestrongest relations that hold between Q and Q*, between Qf and ~Q, betweenQ* and ~β, between Q and ~β, and between Q' and Q*, as recorded in thefollowing results:

(6.1) Q¥Qf iff Q 4 β * iff Q'«f ~Q iff Q*T "Q iff β «f~β iff Q'«+Q*(6.2) Q T Q ' iff β+> β* iff β'+> " β iff β * F " β iff β+> - β iff β'+> β*(6.3) β ~ β ' iff β - Q* iff β ' - ~ β iff β * - " β(6.4) β S β ' iff β ^ β * iff β ' - " β iff β * C " β(6.5) β C β ' iff β->β* iff β'-* - β iff β*S ~β(6.6) β L β ' iff β L β * iff β 'L ~Q iff β*L ~β(6.7) if β L β ' , β C β ' , β S β ' , or Q++Q' then Q- ~Q and β ' - β .

7 Squares of opposition By a square of opposition I mean a display of thestrongest logical relations among cognate quantifiers Qy Q\ β*, and ~Q inwhich the quantifiers are placed at the vertices of a square and their relation-ships are shown along the edges and diagonals, and in which quantifiers that areobverses of one another appear in the same row and quantifiers that are dualsof one another are placed in the same column. I count two squares of opposi-tion as being of the same type if they have the same logical relations along cor-responding edges and diagonals, or if one can be rotated 180° about a verticalor horizontal axis, or about an axis orthogonal to the page, in a way that bringsthis about. Thus, for example, the squares of opposition in Figure 2 are all ofthe same type.

βi C Qί Q2 C Qί Ql S - β 2

IXI IXI IXtQί S -Q, Qϊ S -Q2 Q2 C Q2

Figure 2. Some squares of opposition of the same type.

Using these definitions, and the results given in 6.1-6.7, it follows that thereare exactly four types of square of opposition, as shown in Figure 3. I followtradition by omitting explicit representation of the trivial relation L.

The standard universal quantifier fits the Boolean square of opposition. Ifwe hold to a strict terminology, according to which no modifications of the defi-nitions of the logical relations are to be entertained, then we will not be able tosay (as is often, and correctly, said in a less strict terminology) that, in thepresence of the assumption that the terms involved are nonempty, the universalquantifier and its cognates form a classical square of opposition. To say thatinvolves amending the definition of contrariety, for example, so that the sche-mata (QX)[Y] and (Q'X)[Y] are contraries iff for all nonempty X andY, ~(QX)IY] v ~(Q'X)[Y]Λϊ we disallow such redefinitions, it may at firstseem intuitively unlikely that there will be any quantifiers that fit a classicalsquare. However, as we can determine from Table 3, there are: the quantifier

316 MARK BROWN

Q€9 corresponding to Russellian definite descriptions, i.e., the quantifier 'the',is one! The quantifier 'most', construed in accordance with the semantics offeredfor it at the end of Section 3 above, is another. From Table 3 we can also seethat Qa fits a semi-degenerate square of opposition. In addition, the quantifier'some, but not-all' fits such a square. Finally the quantifier QFand its dual Qτ

fit a degenerate square of opposition, and are the only quantifiers that do.

Type I: Boolean Q Qf

Q* -Q

Type II: Classical Q C Q'

IXIQ* S -Q

Type III: Semi-degenerate Q —— Q'

IXIType IV: Degenerate Q F Q'

TXTQ* T -Q

Figure 3. The four types of square of opposition.

8 Properties of quantifiers We noted in Section 6 that relations betweencognate categorical schemata could be reconstrued as relations between cognatequantifiers. But since cognate quantifiers are inter definable, it follows that theserelations can also be reconstrued as properties of quantifiers. In Table 4 the en-tries (other than Ό' and 'f\ which retain the sense given time in Table 3) aredesignations for the properties Q has by virtue of the fact that the schemata citedabove the entry have the relation cited to its left.

The properties E2-E8 can be given more or less familiar names, followingthe terminology in Bird ([3], p. 55). E2 is the property of obvertibility, E3 partialinvertibility, E4 invertibility, E5 convertibility, E6 obverted convertibility, E7partial contraposability, and E8 contraposability. The properties A3 and A4 arenot normally discussed, let alone named, but might be called partial anti-invertibility and anti-invertibility, respectively, to maintain parallelism withestablished vocabulary. Similarly, the properties S2-S8 arre not commonly dis-cussed, but might be given parallel names by calling S2 obverse subcontrariety,S3 partial inverse subcontrariety, S4 inverse subcontrariety, S5 converse subcon-trariety, S6 obverted converse subcontrariety, S7 partial contrapositive subcon-

GENERALIZED QUANTIFIERS 317

Table 4. Logical Properties of Quantifiers

1 2 3 4 5 6 7 8

A QXY QXY QXY QXY QXY QXY QXY QXY

B QXY QX~Y Q~XY Q~X~Y QYX QY~X Q~YX Q~Y~X

A¥B F F F F F F F F

AΎB T T T T T T T T

A++B El E2 E3 E4 E5 E6 E7 E8

A-B 0 0 A3 A4 0 0 0 0

ASB $ S2 S3 S4 S5 S6 S7 S8

ACB ft C2 C3 C4 C5 C6 C7 C8

ALB ^ L2 L3 L4 L5 L6 L7 L8

trariety, and S8 contrapositive subcontrariety. By replacing 'subcontrariety' by'contrariety', we have names for the properties C2-C8. I will not attempt to pro-vide appropriate names for the vacuous properties L2-L8.

For each of these properties, Table 3 provides a quantifier which exempli-fies that property and exemplifies no stronger property from the same column.The type of square of opposition that a quantifier fits is determined solely byits properties from column 2, but other properties, such as contraposability, arenormally taken to be significant in connection with the square of opposition,and yet are independent of the properties in column 2. For example, Q3 and Qδ

both have the property L2, and neither has any stronger property from column2. It follows that both generate squares of opposition of Type I. But Qδ alsohas the property S5 (converse subcontrariety) while Q3 has no stronger propertyfrom column 5 than L5. Thus we may think of these two quantifiers as gener-ating squares of opposition of different subtypes within the same type. It there-fore becomes of interest to determine into what subtypes squares of oppositionmay fall, based on the other properties quantifiers may possess.

The question what subtypes there are can be answered by answering twoother questions: What combinations of properties may a quantifier have? andHow are the properties of a given quantifier related to those of its cognates? Thislatter question arises because the collection of properties that Q has may deter-mine the collection β ' has, and if in some cases these are different collections,then since Q and Qr fall within the same square of opposition we will want tosay that the two collections of properties determine the same subtype. Three ob-servations simplify the process of answering this second question. First, everyquantifier has the property El trivially. The only question is whether it has oneof the two stronger properties T or F. So we may neglect El.

Second, we can show directly from the definitions of the properties in-volved that Q has a given property P6 from the sixth column iff it has the cor-responding property P7 from the seventh (i.e., Q has E6 iff it has E7, has S6iff it has S7, etc.). Hence one of these two columns may be considered redun-dant, and may be ignored. I choose to ignore column 7. Third, we can show thatQ has a given property P3 from the third column iff it has the correspondingproperty P4 from the fourth. This is so because, by 4.7 and 4.8, all and only

318 MARK BROWN

those combinations of truth values which can be given to (QS)[P] and(Q~S)[P] can be given to (QA)[B] and (Q~A)[~B], where A is S and Bis S s P. Hence we may ignore one of these columns, and I choose to ignorecolumn 3. Elementary use of Rules E, O, D, and C now suffices to give theresults in Table 5.

Table 5. Connections Among Properties of Cognate Quantifiers

Q \Q'Q* -Q Q \Q' Q* -Q Q \Q'Q* -Q Q \Q' Q* -Q

F F T T E4 E4 E4 E4 S5 S8 C8 C5 L6 L6 L6 L6T T F F A4 A4 A4 A4 C5 C8 S8 S5 E8 E5 E5 E8E2 E2 E2 E2 S4 S4 C4 C4 L5 L8 L8 L5 S8 S5 C5 C8S2 S2 C2 C2 C4 C4 S4 S4 E6 E6 E6 E6 C8 C5 S5 S8L2 L2 L2 L2 E5 E8 E8 E5 C6 C6 S6 S6

Thus, for example, if S5(Q), i.e., if Q has the property S5, then S8(Q'),C8(Q*), and C5(~Q). Note too that if (as is the case with the existential quan-tifier) Q is convertible, then ~Q will be too, while Q' and Q* will be contra-posable. That is, it would be impossible to produce a quantifier which wasconvertible but whose dual was not contraposable.

The process of sorting out the various possible subtypes of squares ofopposition is simplified by establishing some elementary results concerning theconsequences of having one or more of the properties E2-E8, A4. There areseveral such results which can be proven using only our six rules. In reportingthese results below I use (for example) Έ5' to abbreviate Έ5(Q)\ so that ineach of these results I am reporting connections among the properties of a singlequantifier. I abbreviate further by letting, for example, T2 iff P4' abbreviateL2 iff L4, and S2 iff S4, and C2 iff C4, and E2 iff E4, and A2 iff A4\

(8.1) If E2: P5 iff P6 iff P8.(8.2) If E4: E2; and

A4 and C5 are impossible.(8.3) If E5: P2 iff P4 iff P6 iff P8;

and A4 is impossible.(8.4) If E6: E2, E4, E5, and E8.(8.5) If E8: P2 iff P4 iff P5 iff P6; and

A4 is impossible.(8.6) If A4: E2, L5, L6, and L8.

To these results we may add, for ease of reference, three results reported previ-ously in connection with Table 5.

(8.7) If not T and not F, then El.(8.8) P6 iff P7.(8.9) P3 iff P4.

One result reported above (in 8.2) is less trivial than the rest to prove,

GENERALIZED QUANTIFIERS 319

namely that if E4(<2) (and nothing stronger) then not C5(Q). Suppose E4(ζ>)and C5(Q), and suppose (QS)[P), for some S and P. By Rule A, (QS)[S&P]9

so because E4(Q) (Q ~ S)[~(S & P ) ] . By Rule A again, (Q~S)[~S &~(S&P)]. By Rule E, (Q~S)[~S]. Since E 4 ( 0 , it follows that (QS)[S].Since C5(Q) we get - (βS)[S] , which is a contradiction. So for any S and P,~(ζλS)[P]. Thus F(Q), contradicting the stipulation that Q have no propertystronger than E4.

One other collection of elementary results is available to simplify furtherthe process of sorting out the various subtypes of square of opposition. Theseresults are all of the following form: if Q has one form of subcontrariety andanother form of contrariety, then it has one or more forms of equivalence prop-erties. These results are all recorded in Table 6, in which the antecedents appearas coordinates, and the consequents as the entries, of the array.

Table 6. Implications of Some Pairs of Properties

C2 C4 C5 C6 C8

S2 . . . E4 E6 E5&E8 E654 E4 . . . E6&E8 E5&E6&E8 E5&E655 E6 E6&E8 . . . E2&E4 E456 E5&E8 E5&E6&E8 E2&E4 . . . E2&E4S8 E6 E5&E6 E4 E2&E4

Let us use expressions like 'SLLSE' to designate combinations of propertiesof a single quantifier —in this case, the combination S2, L4, L5, S6, and E8. Inthe light of 8.7-8.9 we need not give separate mention to the associated prop-erties from columns 1,3, and 7. Thousands of combinations of properties arelexicographically possible, but fewer than 100 such combinations are consistentwith all the results in 8.1-8.9 and Table 6. Of these, some can be associated withone another, using the results in Table 5, as generating the same subtype ofsquare of opposition. For example, if Q has the combination SLSSL, then Q'will have the combination SLLSS, Q* will have the combination CLLCC, and~Q will have the combination CLCCL. We could therefore choose any one ofthese four combinations to represent the subtype of square of opposition whichβ, Q\ Q*, and ~Q fit. It is convenient to stipulate the "alphabetic" order L,C, S, E, A, F, T for the letters involved, and let this generate a lexicographicorder among the combinations. Then when a set of combinations are associatedwith one another by the results in Table 5, as belonging to the same subtype ofsquare, the lexicographically earliest such combination can be taken as repre-sentative of the rest of the set. This procedure puts the subtypes in order by type,and chooses as the representative quantifier, in a set of cognate quantifiers, onewhich can fit in the upper left corner of the square (the A-position, by traditionallabeling) in the standard arrangement of the square given in Figure 3.

When we associate with one another the combinations of properties whichcorrespond to cognate quantifiers, using Table 5, and apply all the constraints

320 MARK BROWN

given in 8.1-8.9 and Table 6, we find that there are only 34 possible subtypesof square of opposition. Examples of quantifiers of each of these subtypes canbe constructed, so there are exactly 34 subtypes. They are given in Table 7.

Table 7. The Subtypes of Square of Opposition

Type I Type II Type III Type IV

LLLLL CLLLL ELLLL ΓFFFF

LLLLC CLLLC ELCCCLLLLE CLLCL ECLLLLLLCL CLLCC ECCCCLLLCC CLCLC EELLLLLCLC CLCCC EEEEELLCCC CCLLL EALLLLCLLL CCLLCLCLLC CCLCLLCLCL CCLCCLCLCC CCCLCLCCLC CCCCCLCCCC CCCCE

9 Rules of immediate inference By a classical rule of immediate inferenceI mean any rule of the form A/.'.B or the form A :: B, where A and B are cog-nate categorical schemata. Every possible classical rule of immediate inferencecorresponds to one or another of the properties of quantifiers introduced inSection 8 and, with the exception of the trivial properties El and L2-L8, eachof the properties of quantifiers corresponds to a rule of immediate inference.Some of these correspondences are evident from our choice of names for theproperties. Thus, for example, any quantifier with the property E8 will permita rule of contraposition. Other correspondences are less obvious. For example,any quantifier Q with the property C6 permits a rule of "conversion by limita-tion" (although the term "limitation" may not be as apt here as it was in thetraditional analysis of categorical propositions), i.e., a rule of the formQXY/.'.Q*YX. Some of the rules of immediate inference that correspond toproperties of quantifiers are entirely unfamiliar. Thus, for example, if Q has theproperty A3 then it permits the rule QXY:: ~Q~ XY.

Once we have determined the collection of properties a quantifier has, andtherefore have determined what properties its cognates have, we will not onlyhave determined what subtype of square of opposition they fall into, but willalso have determined the classical rules of immediate inference they support.Indeed, in determining which subtypes of square of opposition are possible, wehave in effect determined what combinations of classical rules of immediateinference are possible. For each distinct subtype, there is a distinct set of rulesof immediate inference that characterize it. The infinitely many quantifiers ofthe forms 'all but n\ 'with fewer than n exceptions', 'with at most n exceptions',

GENERALIZED QUANTIFIERS 321

'with at least n exceptions', and 'with more than n exceptions' each has the samecollection of properties as 'all'. We therefore know that each generates a squareof opposition of the same subtype (namely LLLLE) as 'all' and that each is sub-ject to exactly the same classical rules of immediate inference. Similarly, theirduals 'exactly n\ 'at least n\ 'more than n\ 'fewer than n\ and 'at most n\respectively, all have the same properties, and therefore support the same clas-sical rules of immediate inference, as 'some'. Thus in a few strokes we obtaincomplete information about the classical rules of immediate inference associatedwith an infinite collection of quantifiers available in natural discourse.

NOTES

1. The formal semantics I will offer is equivalent to the semantics given by Barwise andCooper [2], if we interpret them as intending it to be a defining feature of quanti-fiers (in my sense of Quantifiers') that they denote quantificational functions (as Idefine them in 3.3 below) rather than arbitrary functions from subsets of the domainto families of subsets of the domain. The Barwise and Cooper semantics is adescendent of that given by Montague [8].

2. I am indebted to the referee for the observation that if we are willing to introducedistinguished quantifier constants for universal and existential quantification, and toadd any standard set of rules and axioms for first-order logic, we get a system forwhich the completeness proof has essentially been given in Barwise [1]. The refereeevidently refers to Barwise's sketch of a proof of completeness for indexical quan-tifiers. By dropping from the proof all references to indices, and by modifying theconstruction of quantificational functions in the canonical model so as to conformto the fact that all our quantifiers are relativized, it is indeed possible to adaptBarwise's proof to show the completeness of GQ extended by (or treated as anextension of) first-order logic. A technical problem arises from the fact that, in thecanonical model, we must define the value of a quantificational function even forarguments which do not happen to be the extensions of any formulas with one freevariable. It suffices, however, to let each quantificational function take the empty setas value for each such argument. I second the referee's conjecture that, even withoutdistinguished quantifiers for universal and existential quantification, GQ (understoodto incorporate any appropriate formulation of the propositional calculus) is complete.

REFERENCES

[1] Barwise, J., "Monotone quantifiers and admissible sets," pp. 1-38 in GeneralizedRecursion Theory II, ed., J. E. Fenstad et al., North-Holland, Amsterdam, 1978.

[2] Barwise, J. and R. Cooper, "Generalized quantifiers and natural language,"Linguistics and Philosophy, vol. 4 (1981), 159-219.

[3] Bird, O., Syllogistic and Its Extensions, Prentice-Hall, Englewood Cliffs, NewJersey, 1964.

[4] Copi, I., Introduction to Logic, Macmillan Publishing Co., New York, 1953.

[5] Geach, P. T., Reason and Argument, Basil Blackwell, Oxford, 1976.

322 MARK BROWN

[6] Kahane, H., Logic and Philosophy, Wadsworth Publishing Company, Belmont,California, 1969.

[7] McCawley, J. D., Everything that Linguists have Always Wanted to Know aboutLogic but were ashamed to ask, The University of Chicago Press, Chicago, 1981.

[8] Montague, R., "The proper treatment of quantification in ordinary English,"pp. 247-270 in Formal Philosophy, ed., Richmond H. Thomason, Yale UniversityPress, New Haven, 1974.

[9] Peterson, P. L., "On the logic of 'few', 'many', and 'most'," Notre Dame Journalof Formal Logic, vol. 20 (1979), pp. 155-179.

[10] Peterson, P. L., and R. D. Carnes, "The compleat syllogistic: Including rules andVenn diagrams for five quantities," photocopied typescript, 1981.

[11] Rescher, N. and N. Gallagher, "Venn diagrams for plurative syllogisms,"Philosophical Studies, vol. 16 (1965), pp. 49-55.

[12] Thompson, B., "Syllogisms using 'few', 'many', and 'most'," Notre Dame Journalof Formal Logic, vol. 23 (1982), pp. 75-84.

Department of PhilosophySyracuse UniversitySyracuse, New York 13210